On the Discretization Error of the Discrete Generalized Quantum Master Equation

The transfer tensor method (TTM) [Cerrillo and Cao, Phys. Rev. Lett. 2014, 112, 110401] can be considered a discrete-time formulation of the Nakajima-Zwanzig quantum master equation (NZ-QME) for modeling non-Markovian quantum dynamics. A recent paper [Makri, J. Chem. Theory Comput. 2025, 21, 5037] raised concerns regarding the consistency of the TTM discretization, particularly a spurious term at the initial time $t=0$. This Communication presents a detailed analysis of the discretization structure of TTM, clarifying the origin of the initial-time correction and establishing a consistent relationship between the TTM discrete-time memory kernel $K_N$, and the continuous-time NZ-QME kernel $\mathcal{K}(N\Delta t)$. This relationship is validated numerically using the spin-boson model, demonstrating convergence of reconstructed memory kernels and accurate dynamical evolution as $\Delta t \to 0$. While TTM provides a consistent discretization, we note that alternative schemes are also viable, such as the midpoint derivative/midpoint integral scheme proposed in Makri's work. The relative performance of various schemes for either computing accurate $\mathcal{K}(N\Delta t)$ from exact dynamics, or obtaining accurate dynamics from exact $\mathcal{K}(N\Delta t)$, warrants further investigation.